Work out the equation of this graph (picture included)

First, you have to identify what the graph is: a parabola, a circle, an ellipses, etc. it's a parabola.

to be totally honest, i don't know what the (1, 0) point is for. i thought it was meant to be the vertex, but it doesn't make sense? you find the vertex by calculating the midpoint, if nothing else, and the midpoint here is 0.5 ?? but i chose to disregard it and assume it to be a misprint

the graph gives you the x-intercepts, which are your "zeroes" -- this means that you can format them like so:

y = (x + 3)(x - 4)

this will give you your equation. you just have to list the zeroes as factors, then foil them to get a ax² + bx + c formula:

y = x² - x - 12

however, we generally prefer parabolas to be in vertex form. you can get it there by completing the square:

y = x² - x - 12
move the 12 so that it's with the y
y + 12 = x² - x

now you have to complete the square. this is done by taking your "middle" term on the right side, dividing it by 2, and squaring it:

(-1)/2 = -0.5
(-0.5)^2 = 0.25 or (1/4)

i'm going to write it as a decimal because it's easier to follow. when completing the square, whatever you add to one side, you must add to the other, so:

y + 12 + (0.25) = x² - x + (0.25)
then change the right side of the equation into a squared binomial
y + 12.25 = (x - 0.5)²
subtract 12.25 from each side
y = (x - 0.5)² - 12.25

y = (x - 0.5)² - 12.25 is your formula.